How can a Muslim woman, married to a Muslim man who does not practice Islam and has left her and traveled without divorcing her, obtain a lawful divorce so that she may marry another man?

The questioner's statement, "the husband of this woman does not apply Islam," is general. If it means that he is a Muslim who is negligent in his duties or commits forbidden acts, then it is preferable for his wife to advise him and remind him of God. If he refuses or insists on his negligence, she has the right to seek divorce.

However, if it means that the husband is not committed to Islam at all, such as being a disbeliever who has not adhered to the rulings of Islam after uttering the two testimonies of faith, or an apostate, then her marriage to him is fundamentally void if he married her in this state. She must seek a formal separation from him, and it is not permissible for her to allow him to have conjugal relations with her.

And if the husband was a Muslim and then apostatized, then her marriage to him is suspended pending the waiting period (iddah), and she should not allow him to have conjugal relations with her as long as he remains in his disbelief. If the waiting period ends and he does not return to Islam, then the marriage is dissolved.

It is best to present the full matter to specialized scholars in Islamic centers.